I need to send my gun in to have some work done. This little round part in the middle here under the pin keep backing out during firing and jamming the cylinder release button so I can't open the cylinder.

Can anyone tell me what this round part in the middle is called so when I try to explain the problem, I don't have to call it "round part in the middle under the firing pin"? : ) Thanks.

That part is called the bolt, and it is connected to the cylinder release (screw in cylinder release is bolted to the bolt).
What is likely happening is that the ejector rod is loose, which - when loose - actually lengthens the effective length of the ejector rod, pushing back on the bolt, making it hard to open.
Before looking into replacing the bolt (there is no adjustment at all), I'd look into what is happening to the ejector rod, or it's possible that the center pin (the pin inside the ejector rod) is clogged up or stuck, which will give you the same symptoms.
Open the cylinder and see if the center pin can be pushed forward (center of the ejector star) freely, and make sure the ejector rod is tight (if a late model revolver, it will have left handed theads)

Good luck

__________________
.............SmithNut

That's the cylinder locking bolt. It shouldn't be able to move out unless the actual pin is broken off.

Here's a S&W schematic. The part is part number 58.
The pin on the front sticks through the breech face and locks into the center of the ejector.
Possibly the pin has separated from the rest of the part.

http://www.stevespages.com/ipb-sandw-k38.html